Bremner, Murray R.; Peresi, Luiz A. An application of lattice basis reduction to polynomial identities for algebraic structures. (English) Zbl 1173.17001 Linear Algebra Appl. 430, No. 2-3, 642-659 (2009). In this paper, the authors study computationally how to simplify the polynomial identities describing the trilineat operations classified by the authors in [Commun. Algebra 35, No. 9, 2932–2959 (2007; Zbl 1172.17003)].The authors give a lattice basis reduction to simplify these identities equivalent to the ones obtained considering row canonical forms. To do so, the polynomial identities are represented as the nullspace of a large matrix \(E\) with entries in \(\mathbb Z\) (the expansion matrix). From the row canonical form of \(E\) over \(\mathbb Q\), an integral basis for the nullspace is extracted as a vector space over \(\mathbb Q\). Next they obtain a basis for the nullspace lattice by using the Hermite normal form of the transpose \(E^t\) and a matrix \(U\) such that \(U\cdot E^t=H\). If \(n\) is the dimension of the nullspace of \(E\) over \(\mathbb Q\), the last \(n\) rows of \(U\) form an integral basis for the nullspace lattice.To simplify these basis vectors, the LLL algorithm is used. The paper applies this approach for obtaining simpler polynomial identities for the most difficult case in the classification of trilinear operations. The authors show the pseudo-code of several algorithms applied to compute the lattice basis reduction. Reviewer: Angel F. Tenorio (Seville) Cited in 13 Documents MSC: 17-08 Computational methods for problems pertaining to nonassociative rings and algebras 17A40 Ternary compositions 68W30 Symbolic computation and algebraic computation Keywords:nonassociative algebra; LLL algorithm; Hermite normal form; lattice basis reduction; trilinear operations; polynomial identities Citations:Zbl 1172.17003 × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Adkins, W. A.; Weintraub, S. H., Algebra: An Approach via Module Theory (1992), Springer: Springer Berlin · Zbl 0768.00003 [2] Bremner, M. R.; Hentzel, I. R., Identities for generalized Lie and Jordan products on totally associative triple systems, J. Algebra, 231, 387-405 (2000) · Zbl 0999.17044 [3] Bremner, M. R.; Peresi, L. A., Classification of trilinear operations, Comm. Algebra, 35, 2932-2959 (2007) · Zbl 1172.17003 [4] Cohen, H., A Course in Computational Algebraic Number Theory (1993), Springer: Springer Berlin · Zbl 0786.11071 [5] I.R. Hentzel. Processing identities by group representation. in: Computers in Nonassociative Rings and Algebras (Special Session, 82nd Annual Meeting of the American Mathematical Society, San Antonio, Texas, 1976), Academic Press, New York, 1977, pages 13-40.; I.R. Hentzel. Processing identities by group representation. in: Computers in Nonassociative Rings and Algebras (Special Session, 82nd Annual Meeting of the American Mathematical Society, San Antonio, Texas, 1976), Academic Press, New York, 1977, pages 13-40. [6] Lenstra, A. K.; Lenstra, H. W.; Lovász, L., Factoring polynomials with rational coefficients, Math. Ann., 261, 515-534 (1982) · Zbl 0488.12001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.